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Geometriae Dedicata 93: 1–10, 2002. # 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Uniqueness of Noncompact Spacelike Hypersurfaces of Constant Mean Curvature in Generalized Robertson–Walker Spacetimes Dedicated to the memory of Professor Andre´ Lichnerowicz JOSE´ M. LATORRE and ALFONSO ROMERO Departamento de Geometrıá y Topologıá, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain. e-mail: [email protected] (Received: 12 July 2000; accepted in final form: 27 June 2001) Abstract. On any spacelike hypersurface of constant mean curvature of a Generalized Robertson–Walker spacetime, the hyperbolic angle y between the future-pointing unit normal vector field and the universal time axis is considered. It is assumed that y has a local maximum. A physical consequence of this fact is that relative speeds between normal and comoving observers do not approach the speed of light near the maximum point. By using a development inspired from Bochner’s well-known technique, a uniqueness result for spacelike hypersurfaces of constant mean curvature under this assumption on y, and also assuming certain matter energy conditions hold just at this point, is proved. Mathematics Subject Classifications (2000). Primary 53C42; Secondary 53C50, 53C80. Key words. Bochner–Lichnerowicz’s formula, constant mean curvature, GRW spacetime, spacelike hypersurface.

1. Introduction Spacelike hypersurfaces of constant mean curvature in a spacetime are critical points of the area functional under a certain volume constraint [5] (see also [4]). Such hypersurfaces play an important part in Relativity since it was noted that they can be used as initial hypersurfaces where the constraint equations can be split into a linear system and a nonlinear elliptic equation [8, 12, 14]. A summary of other reasons justifying the study of these hypersurfaces in Relativity can be found in [13]. In this paper, we will consider spacelike hypersurfaces of constant mean curvature in the family of cosmological models called generalized Robertson–Walker (GRW) spacetimes. GRW spacetimes are warped products of a (negatively definite) universal time as base and an arbitrary Riemannian manifold as fiber (see Section 2). This notion was introduced in [1–3] (see also [17, 19] for a systematic study of the geometry of such Lorentzian manifolds). Thus our ambient spacetimes widely extend to those which are classically called Robertson–Walker spacetimes. GRW spacetimes include, for instance, the Einstein–de Sitter spacetime, the Friedmann cosmological models

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JOSE´ M. LATORRE AND ALFONSO ROMERO

and the static Einstein spacetime, as well as other relevant geometric models such as the De Sitter spacetime. On the other hand, small deformations of the metric on the fiber of classical Robertson–Walker spacetimes fit into the class of GRW spacetimes and also conformal changes of the metric of a GRW spacetime with a t-dependent conformal factor produce new GRW ones. Note that a GRW spacetime is not necessarily spatially homogeneous, as in the classical cosmological models case. The spatial homogeneity is, of course, appropriate to consider the universe in the large. However, to consider a more accurate scale this assumption could be not realistic and GRW spacetimes could be suitable spacetimes to model universes with inhomogeneous spacelike geometries [16]. In previous papers [2–4], compact spacelike hypersurfaces of constant mean curvature were studied in these ambient spacetimes. The compacteness assumption is natural if we consider spatially closed GRW spacetimes as cosmological models. Note that the existence of a compact spacelike hypersurface in a GRW spacetime implies that it is spatially closed (see Section 2). Thus, there is no compact spacelike hypersurface in a GRW spacetime not spatially closed (or open, in classical terminology). On the other hand, in the above quoted references the main tool are several Minkowski-type integral formulas which work in the compact case. In this paper, we will study spacelike hypersurfaces of constant mean curvature in (not necessarily spatially closed) GRW spacetimes. As we will comment later, completeness on the hypersurface is not related to our physical setting. So we will adopt a local viewpoint to set up our main results. Our approach uses a distinguished function on the spacelike hypersurface M as a fundamental tool. Namely, the hyperbolic angle y between the future-pointing unit normal vector field N (see the next section for the definition) and a natural unit time the coordinate vector field induced by the like vector field on the GRW spacetime M: In a GRW spaceuniversal time on M, @t (which defines the time-orientation of M). the integral curves of @t are called comoving observers and @t ð pÞ, p 2 M, is time M called an instantaneous comoving observer [18, p. 43]. If p is a point of a spacelike among the instantaneous observers at p, @t ð pÞ and Np appear hypersurface M in M, naturally. So, from the orthogonal decomposition Np ¼ eð pÞ@t ð pÞ þ NFp , we have that cosh yð pÞ coincides with the energy eð pÞ ¼ hNp ; @t ð pÞi that @t ð pÞ meausures for Np . On the other hand, the speed kvð pÞk of the velocity vð pÞ :¼ ð1=eð pÞÞNFp that @t ð pÞ meausures for Np satisfies kvð pÞk2 ¼ tanh2 yð pÞ, [18, pp. 45, 67]. In the special but important case when M is a spacelike slice, i.e. a t ¼ constant hypersurface in a GRW spacetime, we have kvk ¼ 0, i.e y 0. In fact, it is easily seen that this property characterizes such a family of spacelike hypersurfaces. Now consider a compact spacelike hypersurface M in a (necessarily spatially closed) GRW spacetime, then kvk, as a function on M, attains a global maximum, so that kvk do not approach to light speed 1 on M. In this direction, the natural generalization of the compacteness of M would be to assume that sup kvk < 1 holds on all M. However, this natural assumption is much too weak. In order to support this assertion, note that a closed (and, hence, inextendible) spacelike hypersurface does

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UNIQUENESS OF NONCOMPACT SPACELIKE HYPERSURFACES

not satisfy this condition, in general. For instance, the graph in L2 :¼ ðR2 ; dx2 dy2 Þ of a smooth function f : R ! R such that f 0 ðxÞ < 1

if j x j< 1

and

f 0 ðxÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 expð j x jÞ

if j x j 5 1

is clearly a closed subset of L2 , and it is easily seen that sup kvk ¼ 1. Note that this spacelike graph is not complete because its length is finite [10]. On the other hand, the same comment is also true if M is assumed to be geodesically complete. For instance, on the spacelike hyperboloid H2 :¼ fðx; y; zÞ 2 L3 : x2 þ y2 z2 ¼ 1; z > 0g in L3 :¼ ðR3 ; dx2 þ dy2 dz2 Þ we also have sup kvk ¼ 1. A way to control the relative speeds (locally) on M would be to assume that y has a local maximum at some point p0 2 M, which is equivalent to saying that the relative speeds satisfy kvð pÞk 4 kvð p0 Þk < 1 for p near p0 . This will be the assumption we will impose on the spacelike hypersurfaces of constant mean curvature of GRW spacetimes in this paper. Recall that, in any GRW spacetime, every spacelike slice t ¼ t0 is totally umbilic with constant mean curvature H ¼ f 0 ðt0 Þ=fðt0 Þ (see the next section for more details). So, under that assumption on y and under certain matter energy conditions at this point, which have a clear physical meaning, we prove a uniqueness result, Theorem 4.2, which roughly says that under these assumptions a spacelike hypersurface of constant mean curvature is a spacelike slice near the local maximum. In particular, in Corollary 4.4 we generalize, the uniqueness results given by Corollaries 5.3 and 5.4 in [2]. Our technique is completly different from the one in [2]. In fact, our approach works in the noncompact case and, of course, also in the compact one, giving, in particular, new proofs of Corollaries 5.3 and 5.4 in [2]. Our philosophy here follows in spirit the well-known arguments of the so-called Bochner technique (see, for instance, [21]). The fundamental fact in this paper is a differential inequality, Proposition 3.1, which gives a lower estimate of the Laplacian of sinh2 y on the spacelike hypersurface of constant mean curvature M. This inequality is derived from the classical Bochner–Lichnerowicz formula (see Section 3) and takes a good look at the critical points of y, Lemma 4.1. In fact, this is the key result in obtaining our main goal, Theorem 4.2.

2. Preliminaries Let ðF; gÞ be an n-dimensional (connected) Riemannian manifold and let I be an open interval in R endowed with the metric dt2 . Throughout this paper we will denote by the ðn þ 1Þ-dimensional product manifold I F with the Lorentzian metric M h;i ¼ pI ðdt2 Þ þ f 2 ðpI ÞpF ðgÞ;

ð1Þ

where f > 0 is a smooth function on I, and pI and pF denote the projections onto I is a Lorentzian warped product, in the sense of [6] and and F, respectively. That is, M


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[15], with base I, fiber F and warping function f. As introduced in [2], we will refer to as a Generalized Robertson–Walker (GRW ) spacetime. M is said to be Given an n-dimensional manifold M, an immersion x : M ! M given by (1) induces, via x, a Riemannian spacelike if the Lorentzian metric on M metric on M. As usual, the induced metric will be also denoted by h ; i and we will as a spacelike hypersurface M. refer to x : M ! M determines a time-orientation The unitary timelike vector field @t :¼ @=@t 2 XðMÞ Then the time-orientatibily of M allows us to conon the Lorentzian manifold M. ? sider, for each spacelike hypersurface M in M, N 2 X ðMÞ as the only globally defined unitary timelike vector field normal to M in the same time-orientation of @t . From the wrong way Cauchy–Schwarz inequality (see, for instance, [15, Prop. 5.30]), we have hN; @t i 4 1 and hN; @t i ¼ 1 at a point p if and only if Nð pÞ ¼ @t ð pÞ. In fact, hN; @t i ¼ cosh y, where y is the hyperbolic angle, at each point, between the unit timelike vectors N and @t . In what follows, we will refer to y as the hyperbolic angle between M and @t . For any spacelike hypersurface M, we put t :¼ pI x : M ! I its projection on I and x :¼ pF x : M ! F its projection on F. The map x is not only smooth. In fact, it is easy to see that gðdxðXÞ; dxðXÞÞ 5

1 hX; Xi; fðtÞ2

ð2Þ

where fðtÞ :¼ f pI x 2 C1 ðMÞ for any tangent vector X to M and, hence, x is a local diffeomorphism; that is, the spacelike hypersurface and the fiber are locally diffeomorphic. If M is assumed to be geodesically complete and the function fðtÞ is bounded away from 0 on M, then from [9, Lemma 7.3.3] we conclude that x is a covering map and, in particular, an open map. As a consequence, there is no compact spacelike hypersurface in a GRW spacetime with noncompact fiber. It should be noted that x can be also forced to be a covering map from different assumptions. In fact, if M is just edgelessly immersed (e.g. x is a proper map) and if f R does not grow too quickly (specifically, I 1=f is infinite at both ends of the interval I), then x is a covering map [11, Theor. 4.4]. On the other hand, if t ¼ pI x is a constant t0 on M, then this covering map is a local homothety and the equality holds in (2) at t ¼ t0 . As has been used before (see [2]), a spacelike hypersurface such that pI x is a constant t0 , i.e. such that xðMÞ is contained in ft0 g F, will be called a spacelike slice. the timelike vector field K 2 XðMÞ given by Now let us introduce on M K ¼ fðpI Þ@t . This vector field has a nice geometrical property that says, in particular, that it is conformal and this will be the key to obtaining the main tools which follow. In order to decide when the hypersurface is a spacelike slice, we have to see when the hypersurface is orthogonal to @t or, equivalently, orthogonal to K. From the rela and those of the base and the fiber tionship between the Levi-Civita connections of M


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[15, Prop. 7.35], or [6, Chap. 3]), it is not difficult to get H Z K ¼ f 0 ðpI ÞZ;

ð3Þ for any vector field Z on M, where H is the Levi-Civita connection of M. From the Gauss and Weingarten formulas we have the relationship between the and the spacelike hypersurface one, denoted by H, Levi-Civita connection of M H X Y ¼ HX Y hAX; Y iN;

ð4Þ

where X; Y 2 XðMÞ and A is the shape operator associated to N, which is defined by AX :¼ H X N:

ð5Þ

Recall that the mean curvature function corresponding to N is defined to be H :¼ ð1=nÞtrðAÞ. Taking tangential components in (3), we get HY KT þ fðtÞhN; @t iAY ¼ f 0 ðtÞY;

ð6Þ

where KT ¼ fðtÞ@Tt ¼ K þ hK; NiN is the tangential component of K. Note, as an easy consequence of (6), that the shape operator A of the spacelike slice t ¼ t0 satisfies A ¼ f 0 ðt0 Þ=fðt0 Þ I, where I is the identity transformation. Therefore, it is totally umbilic with constant mean curvature H ¼ f 0 ðt0 Þ=fðt0 Þ. Contracting (6), we get fðtÞdivð@Tt Þ þ hHfðtÞ; @Tt i þ fðtÞhN; @t itrðAÞ ¼ nf 0 ðtÞ;

ð7Þ

where div denotes the divergence on M. Now, taking into account Ht ¼ @Tt

ð8Þ

and (7), we get Dt ¼

f 0 ðtÞ fnþ j Ht j2 g nHhN; @t i; fðtÞ

ð9Þ

where Dt :¼ divðHtÞ is the Laplacian of t. are related by the The curvature tensor R of M and the curvature tensor R of M so-called Gauss equation hRðX; Y ÞV; Wi ¼ hRðX; YÞV; Wi þ hAY; WihAX; Vi hAY; VihAX; Wi;

ð10Þ

for any tangent vector fields X; Y; V; W 2 XðMÞ. From (10), we derive the relation between the Ricci tensors Ric of M and Ric of M: RicðX; XÞ ¼ RicðX; XÞ þ hRðX; NÞN; Xi þ hA2 X; Xi þ nHhAX; Xi for all X 2 XðMÞ.

ð11Þ


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and write Now, let Z be a tangent vector to M Z ¼ hZ; @t i@t þ ZF ;

ð12Þ

F

where Z denotes its projection on the fiber. In particular, when Z ¼ N, from (12) we obtain j NF j2 ¼j Ht j2 ¼ sinh2 y, where y is the hyperbolic angle between M and @t . On being a warped product (with a one-dimensional base), it is possithe other hand, M ble to express its curvature in terms of the warping function f and the curvature of the fiber F [15, Props. 7.42, 7.43]. So, we have$ 00 t ; NÞN; @t i ¼ hRð@ t ; NF ÞNF ; @t i ¼ f ðpI Þ j Ht j2 hRð@ fðpI Þ

we get and for the Ricci tensor Ric of M, 00 f ðtÞ f 0 ðtÞ2 F F F þ ðn 1Þ RicðZ; ZÞ ¼ Ric ðZ ; Z Þ þ hZF ; ZF i fðtÞ fðtÞ2 f 00 ðtÞ hZ; @t i2 ; n fðtÞ

ð13Þ

ð14Þ

where RicF denotes the Ricci tensor of the fiber. We end this section recalling important energy conditions which are usually we will say that M obeys the timeassumed on the spacetime. Given a spacetime M, like convergence condition (TCC) if RicðZ; ZÞ 5 0, for all timelike vector Z. It is normally argued that TCC is the mathematical translation that gravity, on average, satisfies the Einstein equations (in the terminology attracts. On the other hand, if M of [18], and, in particular, with zero cosmological constant), then it obeys TCC. From (14) it is easily seen that a GRW spacetime M TCC if and only if its obeys F 00 00 02 warping function satisfies f 4 0 and Ric 5 ðn 1Þ ff f g. is said to have nonvanishing matter fields, or obeys the ubiquitous A spacetime M energy condition [20], if RicðZ; ZÞ > 0, for all timelike vector Z. Note that this energy has nonvanishing matter condition is stronger than TCC and if a GRW spacetime M 00 fields, then f < 0.

3. An Inequality Arising from Bochner–Lichnerowicz’s Formula Recall the well-known Bochner–Lichnerowiz formula [7, p. 83] 1 2 2 Dðj Hu j Þ

¼j Hess u j2 þ RicðHu; HuÞ þhHu; HDui

valid for any u 2 C1 ðMÞ, M a Riemannian manifold with metric h;i and Ricci tensor Ric, and where Hu denotes the gradient of u, Hess u :¼ H2 u is the Hessian tensor of u, j Hess u j2 is its square algebraic trace-norm (i.e. j Hess u j2 :¼ trðHu Hu Þ where Hu denotes the operator defined by hHu ðXÞ; Yi :¼ HessðuÞðX; YÞ, for all X; Y 2 XðMÞ), and Du is the Laplacian of u. Note in this paper we are using the notation RðX; YÞZ ¼ H X H Y Z H Y H X Z H ½X;Y Z for the curvature tensor of M. $


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Now we will apply this formula to the function u ¼ t on the spacelike hypersurface M. By using (6) and (8) we get HX Ht ¼

f 0 ðtÞ X þ hN; @t iAX Xðlog fðtÞÞHt fðtÞ

ð15Þ

for all tangent vector field X. From the last formula we achieve f 0 ðtÞ2 f 0 ðtÞ2 f 0 ðtÞ2 2 2 4 þ hN; @ i trðA Þ þ j Ht j þ2 j Ht j2 þ t fðtÞ2 fðtÞ2 fðtÞ2 f 0 ðtÞ f 0 ðtÞ hN; @t i 2 hN; @t ihAHt; Hti: þ 2nH fðtÞ fðtÞ

j Hess t j2 ¼ n

ð16Þ

On the other hand, from (11), (13) and (14) we obtain RicðHt; HtÞ ¼ ðn 1ÞðlogfÞ00 j Ht j4 þðn 1Þ

f 0 ðtÞ2 j Ht j2 þ fðtÞ2

þ hN; @t i2 RicF ðNF ; NF Þ þ nHhAHt; Hti þ hA2 Ht; Hti:

ð17Þ

A direct computation from (3) gives HhN; @t i ¼ AHt

f 0 ðtÞ hN; @t iHt: fðtÞ

ð18Þ

Now, we assume that H is constant. Under this assumption, from (9) and (18) we get hHt; HDti ¼ ðlog fÞ00 fnþ j Ht j2 g j Ht j2 2 þ nH

f 0 ðtÞ HessðtÞðHt; HtÞ þ fðtÞ

f 0 ðtÞ hN; @t i j Ht j2 nHhAHt; Hti fðtÞ

ð19Þ

Now recall that Schwarz’s inequality gives nH2 4 trðA2 Þ, and equality holds if and only if A is, pointwise, a multiple of the identity transformation, i.e. M is totally Using this fact, from (16), (17) and (19) and taking into account umbilic in M. j Ht j ¼ sinh y, we obtain from Bochner–Lichnerowicz’s formula: be a GRW spacetime and let M be a spacelike hyperPROPOSITION 3.1. Let M The hyperbolic angle y between M and @t surface of constant mean curvature H in M. satisfies the differential inequality 2 1 2 D sinh

f 0 ðtÞ2 f 0 ðtÞ 2 HessðtÞðHt; HtÞ þ þ hA Ht; Hti 2 fðtÞ fðtÞ2 f 0 ðtÞ þ nH2 cosh2 y þ cosh2 y RicF ðNF ; NF Þ 2nH cosh y þ fðtÞ f 0 ðtÞ f 0 ðtÞ þ2 cosh yhAHt; Hti nH cosh y sinh2 y þ fðtÞ fðtÞ

y5n


8 þ ð2n þ 1Þ þ ðn þ 1Þ

f 0 ðtÞ2 f 00 ðtÞ 2 sinh2 y þ sinh y n fðtÞ fðtÞ2

f 0 ðtÞ2 f 00 ðtÞ sinh4 y sinh4 y n 2 fðtÞ fðtÞ

and the equality holds if and only if M is totally umbilic in M.

4. Main Results It should be noticed that the right-hand side of differential inequation obtained in Proposition 3.1 is very complicated and not signed, in general. However, if we restrict our attention only on the critical points of y we get: be a GRW spacetime and let M be a spacelike hypersurface of LEMMA 4.1. Let M If the hyperbolic angle y between M and @t has a constant mean curvature H in M. critical point p0 , then 2 f 0 ðt0 Þ 2 2 1 1 cosh yð p D sinh yð p Þ 5 n 1 þ sinh yð p Þ H Þ 0 0 0 2 2 fðt0 Þ f 00 ðt0 Þ n sinh2 yð p0 Þ cosh2 yð p0 Þ þ cosh2 yð p0 ÞRicFq0 ðNF ; NF Þ; fðt0 Þ p0 Þ. where t0 :¼ tð p0 Þ and q0 :¼ xð Proof. From (18), we have at the critical point p0 , AHtp0 ¼

f 0 ðt0 Þ cosh yð p0 ÞHtp0 fðt0 Þ

ð20Þ

On the other hand, we also have at p0 HessðtÞðHt; HtÞp0 ¼ 0:

ð21Þ

Now, to verify the inequality we use (20) and (21) in the inequality of Proposition 3.1, and throw away some nonnegative terms. & be a GRW spacetime and let x : M ! M be a spacelike THEOREM 4.2. Let M hypersurface of constant mean curvature. Assume that the hyperbolic angle between M and @t attains a local maximun at some point p0 2 M. If either ð1Þ f 00 ðtð p0 ÞÞ < 0, is positive semi-definite at xð p0 Þ. ð2Þ The Ricci tensor of the fiber of M or ð10 Þ f 00 ðtð p0 ÞÞ 4 0, is positive definite at xð p0 Þ, ð20 Þ The Ricci tensor of the fiber of M then, there exists an open neighborhood U of p0 in M which is a spacelike slice.

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Proof. From Lemma 4.1, we have 2 1 2 D sinh

yð p0 Þ 5 n

f 00 ðt0 Þ sinh2 yð p0 Þ cosh2 yð p0 Þ þ fðt0 Þ

þ cosh2 yð p0 Þ RicFq0 ðNF ; NF Þ

ð22Þ

and now we know D sinh2 yð p0 Þ 4 0: Consider now the first couple of assumptions. Using taking into account (23), we obtain

ð23Þ RicFq0 ðNF ; NF Þ 5 0

in (22) and

f 00 ðt0 Þ sinh2 yð p0 Þ ¼ 0; but we are also assuming f 00 ðt0 Þ < 0, which gives yð p0 Þ ¼ 0. Finally, we know that yð pÞ 4 yð p0 Þ on some open neighborhood U of p0 . Therefore, yð pÞ ¼ 0, for all p in U, which means that U is a spacelike slice. Next, consider the second couple of assumptions. From f 00 ðt0 Þ 4 0 we get RicFq0 ðNF ; NF Þ ¼ 0; but now we know that RicFq0 is positive definite, and therefore NF ¼ 0 at this point. We end the proof recalling that j NF j2 ¼ sinh2 y, which implies again yð p0 Þ ¼ 0. As immediate consequences we get the following corollaries: COROLLARY 4.3. The only analytical spacelike hypersurfaces of constant mean curvature in a GRW spacetime, whose hyperbolic angle with @t attains a local maximum, where one of the two couple of assumptions of Theorem 4:2 are fulfilled, are open subsets of spacelike slices. COROLLARY 4.4. The only spacelike hypersurfaces of constant mean curvature in a GRW spacetime, whose hyperbolic angle with @t attains a global maximum, where one of the two couple of assumptions of Theorem 4:2 are fulfilled, are open subsets of spacelike slices. Note that this result widely extends and give a new proof of [2, Cors. 5.3, 5.4]. To end this paper, we come back to the Physical motivation in Section 1. So, we have obtained: In a GRW spacetime obeying the ubiquitous energy condition, the only spacelike hypersurfaces of constant mean curvature possessing a global maximun of the observed speed by the comoving observers are open subsets of spacelike slices.

Acknowledgements The authors would like to thank Miguel Sańchez for his clarifying comments on the physical interpretation of geometric assumptions in this paper. We also thank the referee for his criticism and valuable suggestions. Partially supported by a MCYTFEDER Grant BFM2001-2871-C04-01.

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